Stochastic reaction timings that lead to Poisson-distributed counts
description
Transcript of Stochastic reaction timings that lead to Poisson-distributed counts
![Page 1: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/1.jpg)
𝑡
Stochastic reaction timings that lead to Poisson-distributed counts
1
Stochastic transcription with stochastic degradation
Stochastic transcription with deterministic degradation
𝑡
![Page 2: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/2.jpg)
Many (usually unproductive) attempts at mRNA transcription
2
𝑡
1 transcription event
many unproductive attempts=
1 spin represents bunch of attempts
tSURVIVE
tCOUNTtSURVIVEPoisson-distributed # transcriptions during
Poisson-distributed copy #of mRNA at
+
![Page 3: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/3.jpg)
3
𝑡
Stochastic transcription with stochastic degradation
Stochastic transcription with deterministic degradation
𝑡
Stochastic reaction timings that lead to Poisson-distributed counts
![Page 4: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/4.jpg)
Combine stochastic transcription with stochastic degradation
4
1 transcription event
many unproductive attempts=
1 spin represents bunch of attempts
Survives many attempts at degradation
tSURVIVE
![Page 5: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/5.jpg)
Transcription as a Poisson process
5
𝑡
tCOUNT
![Page 6: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/6.jpg)
Exponential distribution of survival times
6
𝑡
tCOUNTtSOURCE
![Page 7: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/7.jpg)
Exponential distribution of survival times
7
𝑡
tCOUNTtSOURCE
![Page 8: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/8.jpg)
Probability of survival illustrated by reverse exponential decay
8
𝑡
tCOUNTtSOURCE
![Page 9: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/9.jpg)
Probability of survival illustrated by reverse exponential decay
9
𝑡
tCOUNTtSOURCE
![Page 10: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/10.jpg)
Probability of survival illustrated by reverse exponential decay
10
𝑡
tCOUNTtSOURCEtEARLIER
![Page 11: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/11.jpg)
Probability of survival illustrated by reverse exponential decay
11
𝑡
tCOUNTtSOURCEtLATER
![Page 12: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/12.jpg)
Probability of survival illustrated by reverse exponential decay
12
𝑡
Tran
scrip
tion
Surv
ival
tCOUNT
![Page 13: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/13.jpg)
Prob. transcribed x Prob. Survived = Prob. counted
13
Tran
scrip
tion
Surv
ival
Coun
ted
X=
pTRANSCR = 1/20
pSURVIVE = 1/3
pCOUNT tCOUNT
𝑡
![Page 14: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/14.jpg)
Multiple “inefficient” wheels look like single “efficient” wheel
14
𝑡
Tran
scrip
tion
Surv
ival
Coun
ted
X=
≈
ABC2 C1D1D2D3E1E2
Copies of A:
E3E4E5E6E7E8E9F1
![Page 15: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/15.jpg)
𝑡
ABC2 C1D1D2D3E1E2E3E4E5E6E7E8E9F1
15
Tran
scrip
tion
Surv
ival
Coun
ted
X=
≈
Yellow icingon
blue cake
Copies of A:Cannot make 6th copy of A
Not enoughfrosting
. . .
Finite number of “effective” wheels
![Page 16: Stochastic reaction timings that lead to Poisson-distributed counts](https://reader036.fdocuments.in/reader036/viewer/2022062810/56815ab8550346895dc87356/html5/thumbnails/16.jpg)
Same statistics for wheels uneven and even in time
16
≈
𝑡
𝑡Stochastic transcription with stochastic degradation
Stochastic transcription with deterministic degradation
Poisson-distributed number of eventsassociated with passing of time
Poisson-distributedinstantaneous copy number
+